Let $n$ be an integer with prime factorization $n = p_1^{n_1} p_2^{n_2} ... p_k^{n_k}$ , where the $p_i$ are distinct primes. How many positive divisors does the integer $n$ have?
Here is what I have:
$p_1^{n_1}$ has the following divisors: $1, p_1, {p_1}^2, {p_1}^3, ... p_1^{n_1}$
So that is $(n_1 + 1)$ divisors
And for $p_2^{n_2}$ it would be $(n_2 + 1)$, and so on.
So n has $(n_1 + 1)(n_2 + 1)(n_3 +1)... (n_j + 1)$ divisors.
Is this correct?